A review of Lie 2-algebras

Abstract

We first recall two equivalent definitions of Lie $2$-algebras, categorification of Lie algebras and $2$-term $L_\infty$-algebras. Then we present four different kinds of Lie $2$-algebras from $2$-plectic manifolds, Courant algebroids, homotopy Poisson manifolds and affine multivector fields on a Lie groupoid respectively. Moreover, we recall the cohomology theory of Lie $2$-algebras and analyze its lower degree cases. The integration of strict Lie $2$-algebras to strict Lie $2$-groups is also discussed.